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Page 1: Repeated Measures ANOVA Quantitative Methods in HPELS 440:210.

Repeated Measures ANOVA

Quantitative Methods in HPELS

440:210

Page 2: Repeated Measures ANOVA Quantitative Methods in HPELS 440:210.

Agenda

Introduction The Repeated Measures ANOVA Hypothesis Tests with Repeated Measures

ANOVA Post Hoc Analysis Instat Assumptions

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Introduction Recall There are two possible

scenarios when obtaining two sets of data for comparison: Independent samples: The data in the first

sample is completely INDEPENDENT from the data in the second sample.

Dependent/Related samples: The two sets of data are DEPENDENT on one another. There is a relationship between the two sets of data.

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Introduction Three or more data sets?

If the three or more sets of data are independent of one another Analysis of Variance (ANOVA)

If the three or more sets of data are dependent on one another Repeated Measures ANOVA

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Agenda

Introduction The Repeated Measures ANOVA Hypothesis Tests with Repeated Measures

ANOVA Post Hoc Analysis Instat Assumptions

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Repeated Measures ANOVA Statistical Notation Recall for ANOVA:

k = number of treatment conditions (levels)nx = number of samples per treatment level

N = total number of samples N = kn if sample sizes are equal

Tx = X for any given treatment level

G = TMS = mean square = variance

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Repeated Measures ANOVA

Additional Statistical Notation:P = total score for each subject (personal

total)Example: If a subject was assessed three

times and had scores of 3, 4, 5 P = 12

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Repeated Measures ANOVA

Formula Considerations Recall for ANOVA:SSbetween = T2/n – G2/N

SSwithin = SSinside each treatment

SStotal = SSwithin + SSbetween

SStotal = X2 – G2/N

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ANOVA Formula Considerations:

dftotal = N – 1

dfbetween = k – 1

dfwithin = (n – 1) dfwithin = dfin each treatment

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ANOVA Formula Considerations:

MSbetween = s2between = SSbetween / dfbetween

MSwithin = s2within = SSwithin / dfwithin

F = MSbetween / MSwithin

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Repeated Measures ANOVA

New Formula Considerations:SSbetween SSbetween treatments = T2/n – G2/N

SSbetween subjects = P2/k – G2/N

SSwithin SSwithin treatments = SSinside each treatment

SSerror = SSwithin treatments – SSbetween subjects

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Repeated Measures ANOVA

New Formula Considerations:dfbetween dfbetween treatments = k – 1

dfwithin dfwithin treatments = N – k

dfbetween subjects = n – 1

dferror = (N – k) – (n – 1)

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Repeated Measures ANOVA

MSbetween treatments=SSbetween treatments/dfbetween treatments

MSerror = SSerror / dferror

F = MSbetween treatments / MSerror

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Repeated Samples Designs One-group pretest posttest (repeated

measures) design: Perform pretest on all subjects Administer treatments followed by posttests Compare pretest to posttest scores and posttest to

posttest scores

O X O X O

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Agenda

Introduction The Repeated Measures ANOVA Hypothesis Tests with Repeated Measures

ANOVA Post Hoc Analysis Instat Assumptions

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Hypothesis Test: Repeated Measuers ANOVA Example 14.1 (p 457) Overview:

Researchers are interested in a behavior modification technique on outbursts in unruly children

Four students (n=4) are pretested on the # of outbursts during the course of one day

Teachers begin using “cost-response” technique

Students are posttested one week later, one month later and 6 months later

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Hypothesis Test: ANOVA

Questions:What is the experimental design?What is the independent variable/factor? How many levels are there?What is the dependent variable?

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Step 1: State Hypotheses

Non-Directional

H0: µpre = µ1week = µ1month = µ6months

H1: At least one mean is different than the others

Step 2: Set Criteria

Alpha () = 0.05

Critical Value:

Use F Distribution Table

Appendix B.4 (p 693)

Information Needed:

dfbetween treatments = k – 1 = 4 – 1 = 3

dferror = (N-k)-(n-1) = (16-4)-(4-1) = 9

Table B.4 (p 693)

Critical value = 3.86

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Step 3: Collect Data and Calculate Statistic

Total Sum of Squares

SStotal = X2 – G2/N

SStotal = 222 – 442/20

SStotal = 222 - 121

SStotal = 101

Sum of Squares Between each Treatment

SSbetween treatment = T2/n – G2/N

SSbetween treatment = 262/4+82/4+62/4+42/4 – 442/20

SSbetween treatment = (169+16+9+4) - 121

SSbetween treatment = 77

Sum of Squares Within each Treatment

SSwithin = SSinside each treatment

SSwithin = 11+2+9+2

SSwithin = 24

Sum of Squares Between each Subject

SSbetween subjects = P2/k – G2/N

SSbetween subjects = (122/4+62/4+102/4+162/4) - 442/16

SSbetween subjects = (36+9+25+64) – 121

SSbetween subjects = 13

Sum of Squares Error

SSerror = SSwithin treatments – SSbetween subjects

SSerror = 24 - 13

SSwithin = 11

Raw data can be found in Table14.3 (p 457)

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Step 3: Collect Data and Calculate Statistic

Mean Square Between each Treatment

MSbetween treatment = SSbetween treatment / dfbetween treatment

MSbetween treatment = 77 / 3

MSbetween = 25.67

Mean Square Error

MSerrorn = SSerror / dferror

MSerror = 11 / 9

MSwithin = 1.22

F-Ratio

F = MSbetween treatment / MSerror

F = 25.67 / 1.22

F = 21.04

Step 4: Make Decision

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Agenda

Introduction Repeated Measures ANOVA Hypothesis Tests with Repeated Measures

ANOVA Post Hoc Analysis Instat Assumptions

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Post Hoc Analysis What ANOVA tells us:

Rejection of the H0 tells you that there is a

high PROBABILITY that AT LEAST ONE difference exists SOMEWHERE

What ANOVA doesn’t tell us:Where the differences lie

Post hoc analysis is needed to determine which mean(s) is(are) different

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Post Hoc Analysis

Post Hoc Tests: Additional hypothesis tests performed after a significant ANOVA test to determine where the differences lie.

Post hoc analysis IS NOT PERFORMED unless the initial ANOVA H0 was rejected!

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Post Hoc Analysis Type I Error Type I error: Rejection of a true H0

Pairwise comparisons: Multiple post hoc tests comparing the means of all “pairwise combinations”

Problem: Each post hoc hypothesis test has chance of type I error

As multiple tests are performed, the chance of type I error accumulates

Experimentwise alpha level: Overall probability of type I error that accumulates over a series of pairwise post hoc hypothesis tests

How is this accumulation of type I error controlled?

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Two Methods Bonferonni or Dunn’s Method:

Perform multiple t-tests of desired comparisons or contrasts

Make decision relative to / # of testsThis reduction of alpha will control for the

inflation of type I error Specific post hoc tests:

Note: There are many different post hoc tests that can be used

Our book only covers two (Tukey and Scheffe)

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Repeated Measures ANOVA Bonferronni/Dunn’s method is appropriate

with following consideration:Use related-samples T-tests

Tukey’s and Scheffe is appropriate with following considerations:Replace MSwithin with MSerror in all formulasReplace dfwithin with dferror in all formulas

Note: Statisticians are not in agreement with post hoc analysis for Repeated Measures ANOVA

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Agenda

Introduction The Repeated Measures ANOVA Hypothesis Tests with Repeated

Measures ANOVA Post Hoc Analysis Instat Assumptions

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Instat Label three columns as follows:

Block: This groups your data by each subject.

Example: If you conducted a pretest and 2 posttests (3 total) on 5 subjects, the block column will look like:

1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 Treatment: This tells you which treatment

level/condition occurred for each data point. Example: If each subject (n=5) received three

treatments, the treatment column will look like: 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

Response: The data for each subject and treatment condition

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Instat Convert the “Block” and “Treatment”

columns into “factors”: Choose “Manage”

Choose “Column Properties” Choose “Factor” Select the appropriate column to be converted Indicate the number of levels in the factor Example: Block (5 levels, n = 5), Treatment (3

levels, k = 3) Click OK

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Instat

Choose “Statistics” Choose “Analysis of Variance”

Choose “General” Response variable:

Choose the Response variable Treatment factor:

Choose the Treatment variable Blocking factor:

Choose the Block variable Click OK. Interpret the p-value!!!

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Instat

Post hoc analysis: Perform multiple related samples t-Tests

with the Bonferonni/Dunn correction method

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Reporting ANOVA Results Information to include:

Value of the F statistic Degrees of freedom:

Between treatments: k – 1 Error: (N – k) – (n – 1)

p-value Examples:

A significant treatment effect was observed (F(3, 9) = 21.03, p = 0.002)

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Reporting ANOVA Results An ANOVA summary table is often

included

Source SS df MS

Between 77 3 25.67 F = 21.03

Within Treatments 24 12

Between subjects 13 3

Error 11 9 1.22

Total 101 15

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Agenda

Introduction The Analysis of Variance (ANOVA) Hypothesis Tests with ANOVA Post Hoc Analysis Instat Assumptions

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Assumptions of ANOVA Independent Observations Normal Distribution Scale of Measurement

Interval or ratio Equal variances (homogeneity) Equal covariances (sphericity)

If violated a penalty is incurred

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Violation of Assumptions Nonparametric Version Friedman Test

(Not covered) When to use the Friedman Test:

Related-samples design with three or more groups

Scale of measurement assumption violation: Ordinal data

Normality assumption violation: Regardless of scale of measurement

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Textbook Assignment

Problems: 5, 7, 10, 23 (with post hoc)